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This is happening due to permissions on the 'Network Service' folder that SQL server uses for Linked Server setup. Inorder to access the linked server using another account (assuming that you have the server permissions set correctly) you will need to do the following:

Open Windows Explorer and navigate to the following folder (follow the next steps carefully as you are now dealing with system files) ‘C:\Windows\ServiceProfiles\NetworkService\AppData\Local’. This directory contains a folder called ‘Temp’, right click on this and select ‘Properties’. In the following dialog click the ‘Security’ tab. If you can see the Standard User account that you want to setup to use the Access Database Engine, this means that it already has some privileges on this folder, if it is not in the ‘Group of user names:’ list, it needs to be added. Either way, click the ‘Edit…’ button underneath the ‘Group of user names:’ list, this will launch the ‘Permissions for Temp’ dialog. If you need to add the Standard User to the permissions group do so using the ‘Add..’ button, making sure the account name is spelt correctly in the following ‘Select users or Groups’ dialog. Once the Standard User is on the permissions list, select that user and in the ‘Permissions for Standard User’ selection-menu check the ‘Full Control’ box under the ‘Allow’ column. Click OK; click OK again to exit this menu.

You should now be able to run access queries as any user that you provide permissions for in this way.

You will also have to make sure that you have run the following queries in Management Studio to correctly configure the Access Engine